1. Complementary Mathematics Various Degrees of the Number ’ s Distinction Doron Shadmi, Moshe Klein Gan-Adam Ltd.

2. Hilbert ’ s 24 th challenge: In the end of his famous ICM 1900 lecture Prof. David Hilbert was concern about the possibility that Mathematics will be split up into separate branches that do not communicate with each other, and he said: “ I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts ”

3. Complementary Mathematics: This work is motivated by Hilbert ’ s organic paradigm, and our solution is based on the relations between the local and the non-local, as observed among 5 years old children that were asked to describe the relations between a point and a line.

4. 5 years old children observation 1: Nevo:"We imagine some shape in our mind, and then we draw it. But we can draw shapes without lines and points and use them in order to create new shapes; all these shapes are in our imagination." Nevo positions his source of creativity in his mind; he discovers that an abstract idea is independent on any particular representation. Sivan: "The shape is in our imagination; we can think about it, but it does not exist. So, we draw it, because we can see it within our hearts." Sivan is aware of the difference between potential and existing things. In his experience, the innovation emerges from the heart.

5. 5 years old children observation 2: Tohar:" If a point and a line are not friends and they are not reproduce, then they will remain few and nothing will be born." Tohar is aware of the distinction between a point and a line. He realizes that new drawings can be created by associating them. Ofri: "The point tries to catch the line, but it cannot catch it because the line is too high." Ofri understands that a line has a property (height) unreachable by a point.

6. Membership, two options: Element is either a set or an urelement. Sub-element is an element that defines another element. Option 1: A membership between an element and its sub-elements (notated by є ). Option 2: A membership between an element and other elements, which are not necessarily its sub- elements (notated by €, where ₡ is “ not a member ” ). Bridging = An option 2 membership = € Element is either a set or an urelement. Sub-element is an element that defines another element. Option 1: A membership between an element and its sub-elements (notated by є ). Option 2: A membership between an element and other elements, which are not necessarily its sub- elements (notated by €, where ₡ is “ not a member ” ). Bridging = An option 2 membership = €

7. Locality and non-locality: x and A are placeholders of an element. Definition 1: If x € A xor x ₡ A then x is local. Definition 2: If x € A and x ₡ A then x is non-local. is a local member (if x= and A=__ then __ xor __ ) A set ’ s member is a local member ( x € A xor x ₡ A ) __ is a non-local member (if x= __ and A= then __ and __ ).......

8. Bridging and Symmetry: A bridging is measured by symmetrical states that exist between local elements and a non-local urelement. No bridging (nothing to be measured) A single bridging (a broken-symmetry, notated by ) More than a single bridging that is measured by several symmetrical states, which exist between parallel symmetry (notated by ) and serial broken-symmetry (notated by ).

9. Bridging and Modern Math: Most of modern mathematical frameworks are based only on broken- symmetry (marked by white rectangles) as a first-order property. We expand the research to both parallel and serial first-order symmetrical states in one organic meta-framework, based on the bridging between the local and the non-local.

10. Organic Natural Numbers: Armed with symmetry as a first-order property, we define a bridging that cannot be both a cardinal and an ordinal (represented by each one of the magenta patterns). The products of the bridging between the local and the non-local are called Organic Natural Numbers.

11. Organic Natural Numbers 4 and 5:

12. Complementary relations between multiplication and addition (+1) (1*2) (1*3)((1*2)+1)(((+1)+1)+1) ((+1)+1)

13. Locality, non-locality and the Real-line: If we define the Real-line as a non-local urelement, then no set is a continuum. By studying locality and non-locality along the real-line we discovered a new kind of numbers, the non-local numbers. For example: The diagram above is a proof without words that 0.111 … is not a base 2 representation of number 1, but the non-local number 0.111 … < 1. The exact location of a non-local number does not exit.

14. Non-local numbers: One can ask: "In that case, what number exists between 0.111 … [base 2] and 1?". The answer is "Any given base n>2 (k=n-1) non-local number 0.kkk … ", for example:

15. Non-locality and Infinity: If the real-line is a non-local urelement, then Cantor ’ s second diagonal is a proof of the incompleteness of R set, when it is compared to the real-line: { {{ },{ },{ },{ },{ },...} {{x},{ },{ },{x},{ },...} {{ },{x},{x},{ },{ },...} {{x},{x},{ },{x},{x},...} {{ },{ },{x},{ },{ },...}... } The non-finite complemntary multiset {{x},{x},{},{},{x},…} is added to the non-finite set of non-finite multisets, etc., etc. … ad infinitum, and R completeness is not satisfied. The non-finite complemntary multiset {{x},{x},{},{},{x},…} is added to the non-finite set of non-finite multisets, etc., etc. … ad infinitum, and R completeness is not satisfied.

16. Incompleteness and proportion A={1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12, … } B={2, 4, 6, 8,10,12, … } If some member is in B set, then it also in A set and since 12 is in B set, in this particular example, then it is also in A set, and the accurate 1-1 mapping is: A={1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12, … } ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ B={2, 4, 6, 8,10,12, … } etc., … etc. ad infinitum B={2, 4, 6, 8,10,12, … } etc., … etc. ad infinitum. Sets A and B are non-finite sets; applying the proportion concept, any non-finite set is no more than a potential-infinity, while distinguishing between locality and non-locality.. Sets A and B are non-finite sets; applying the proportion concept, any non-finite set is no more than a potential-infinity, while distinguishing between locality and non-locality..

17. A new non-finite arithmetic Let @ be a cardinal of a non-finite set, and let C and D be non-finite sets. If |C| = @ and |D| = @-2^@, then |C| > |D| by 2^@. By using the new notion of the Non-finite, we have both non-finite sets and an arithmetic between non-finite sets, which its results are non-finite sets, for example: By Cantor א 0 = א 0 +1, by the new notion @+1 > @. By Cantor א 0 א 0 and א 0 -1 is problematic. By the new notion 3^@ > 2^@ > @ > @-1 etc. By using the new notion of the Non-finite, both cardinals and ordinals are commutative because of the inherent incompleteness of any non- finite set. In other words, @ is used for both ordered/unordered non- finite sets and x+@ = @+x in both cases.

18. A further research We believe that further research into various degrees of the number's distinction (measured by symmetry and based on bridging between locality and non locality) is the right way to fulfill Hilbert's organic paradigm of the mathematical language. Finally, research by Dr. Linda Kreger Silverman over the last two decades demonstrates that there are two kinds of learners: Auditory-Sequential Learners (ASL) and Visual-Spatial-Learners (VSL). Complementary mathematics is a model that bridges between ASL and VSL. Thank you